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The electromagnetoacoustic surface wave in a piezoelectric medium: TheBleustein–Gulyaev modeShaofan Li Citation: Journal of Applied Physics 80, 5264 (1996); doi: 10.1063/1.363466 View online: http://dx.doi.org/10.1063/1.363466 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/80/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Beam propagation algorithms for frequency response of surface acoustic wave directional couplers J. Appl. Phys. 80, 6614 (1996); 10.1063/1.363784 Propagation of surface waves in a prestressed piezoelectric material J. Acoust. Soc. Am. 100, 2112 (1996); 10.1121/1.417921 Single mode Lamb wave excitation in thin plates by Hertzian contacts Appl. Phys. Lett. 69, 146 (1996); 10.1063/1.116902 Application of exceptional wave theory to materials used in surface acoustic wave devices J. Appl. Phys. 79, 8936 (1996); 10.1063/1.362624 Firstorder velocity shift and reflection coefficient for surface acoustic wave propagation J. Appl. Phys. 79, 8230 (1996); 10.1063/1.362463
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The electromagneto-acoustic surface wave in a piezoelectric medium:The Bleustein–Gulyaev mode
Shaofan Lia)Department of Mechanical Engineering, Technological Institute, 2145 Sheridan Road,Northwestern University, Evanston, Illinois 60208
~Received 20 June 1996; accepted for publication 30 July 1996!
By discarding the quasi-static approximation, this paper gives the exact solutions of a shearhorizontal electromagneto-acoustic surface wave mode in a class of piezoelectric media. As thewave speed is much less than the speed of light, the solution degenerates to the well-knownBleustein–Gulyaev wave, or Maerfeld–Tournois wave. Taking into account both optical effect aswell as the contribution from the rotational part of electric field, the solutions obtained here are notonly valid for any wave speed range, but also provide accurate formulas to evaluate theacousto-optic interaction due to the piezoelectricity. ©1996 American Institute of Physics.@S0021-8979~96!06121-X#
I. INTRODUCTION
About thirty years ago, Bleustein1 and Gulyaev2 simul-taneously discovered that there exists a shear horizontal~SH!electro-acoustic surface mode in a class of transversely iso-tropic piezoelectric media, which is known today as theBleustein–Gulyaev wave. The Bleustein–Gulyaev~BG! sur-face wave is an unique result in the repertoire of surfaceacoustic wave~SAW! theory, because it has no counterpartin purely elastic solids. As a matter of fact, since then, theBG wave theory has become one of the cornerstones for themodern electro-acoustic technology.
The BG wave is essentially a coupled surface wave be-tween the acoustic mode and the soft ferroelectric mode; inother words, the quasi-static approximation is adopted for theelectromagnetic field. Under this assumption, both the opti-cal effect as well as the contribution from the rotational partof electric field are neglected. Although it is generally be-lieved that the optical effect is minor, it is certainly of prac-tical interest to accurately predict the piezoelectricity-induced electromagnetic radiation, which might be helpful insome engineering applications, such as optical detection, aswell as nondestructive evaluation in general. In this paper, adetailed account is given of the coupled SH electromagneto-acoustic surface wave in a transversely isotropic piezoelec-tric medium.
In early studies, besides technical considerations, themain reason for adopting the quasi-static approximationmight have been, perhaps, a psychological one. The percep-tion was that solving a fully-coupled Maxwell–Christoffelequation might be too involved, or too complicated to obtainany meaningful results in physics. In this paper, it has beenshown, on the contrary, that there exist some remarkablesimple velocity equations for the fully-coupled SHelectromagneto-acoustic surface wave.
II. FORMULATION
By adopting the notations in Auld,3,4 the governingequations of the problem are listed as follows.
~i! Maxwell’s equations
2¹3E5]B
]t, ~1!
¹3H5]D
]t, ~2!
whereE,B, andH are the electric field, magnetic induction,and magnetic field respectively;
~ii ! Equations of motion
¹•s5r]2u
]t22F, ~3!
wheres is the stress tensor,u is the displacement vector,andF is the body force;
~iii ! The constitutive equations
B5m0H, ~4!
D5es•E1e:«, ~5!
s52e•E1cE:«, ~6!
wherees,e, andcE are the specific dielectric tensor, piezo-electric constant tensor, and elastic stiffness constant tensorrespectively, andm0 is the magnetic permeability constant inthe vacuum.
In the constitutive equations~5! and~6!, the strain tensor« is defined as
«:5 12 @¹u1~¹u!T#5:¹su. ~7!
Letting F50, one may derive the following fully-coupled Maxwell–Christoffel equations@Auld4 ~8.105! and~8.106!# by proper manipulations,
¹•cE:¹su5r]2u
]t21¹•~e•E!, ~8!
2¹3¹3E5m0es•
]2E
]t21m0e:¹s
]2u
]t2. ~9!
a!Electronic mail: [email protected]
5264 J. Appl. Phys. 80 (9), 1 November 1996 0021-8979/96/80(9)/5264/6/$10.00 © 1996 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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Consider the coupling between the anti-plane acousticmode and the in-plane electromagnetic mode, i.e.,
u5@0,0,w~x1 ,x2 ,t !# ~10!
E5@E1~x1 ,x2 ,t !,E2~x1 ,x2 ,t !,0#. ~11!
The coupled wave equations~8! and ~9! can be simplifieddrastically. For the hexagonal~6 mm! piezoelectric material,equations~8! and ~9! reduce to
c44E ¹2w5r
]2w
]t21e15¹•E, ~12!
2¹3¹3E5m0e11s ]2E
]t21m0e15¹
]2w
]t2, ~13!
where the differential operator¹ and the electric fieldE areredefined as two-dimensional vectors, i.e.,
¹:5 i]
]x11 j
]
]x2~14!
E:5E1i1E2j . ~15!
Let
E52¹f21
cl
]A
]t, ~16!
wheref andA are the scalar potential and the vector poten-tial respectively, and the constant,cl :5(m0e11
s )21/2, is thespeed of light in the piezoelectric material. The decomposi-tion ~16! can be uniquely determined by imposing the fol-lowing Lorentz gauge constraint within the transversely iso-tropic plane,
¹•A11
cl
]f
]t50. ~17!
Subsequently, the coupled wave equations~12! and~13!can be further separated into two groups, namely, the purelyelectro-acoustic wave equations,
5 c44E ¹2w2r
]2w
]t25 2e15S ¹2f2
1
cl
]2f
]t2 D ,e15e11s ¹2w 5 S ¹2f2
1
cl
]2f
]t2 D ;~18!
and the rotational part of electromagneto-acoustic waveequations,
5 ¹2A121
cl2
]2A1
]t25 2m0e15cl
]2w
]t]x1,
¹2A221
cl2
]2A2
]t25 2m0e15cl
]2w
]t]x2.
~19!
Define
c44:5c44E 1
e152
e11s , ~20!
ca :5S c44r D 1/2, ~21!
f :5cl
2
cl22ca
2 ; ~22!
and introduce a new scalar potential functionc,
c:5f2e15e11s fw. ~23!
The purely electro-acoustic wave equations~18! can becompletely decoupled as follows,
5 ¹2w21
ca2
]2w
]t25 0,
¹2c21
cl2
]2c
]t25 0.
~24!
Accordingly, the relevant constitutive equations take thefollowing forms,
s135 c44]w
]x11e15
]c
]x11e15cl
]A1
]t, ~25!
s235 c44]w
]x21e15
]c
]x21e15cl
]A2
]t, ~26!
D15e15~12 f !]w
]x12e11
s ]c
]x12
e11s
cl
]A1
]t, ~27!
D25e15~12 f !]w
]x22e11
s ]c
]x22
e11s
cl
]A2
]t, ~28!
wherec44:5c44E 1 f e15
2/e11s .
III. SOLUTIONS
In what follows, the SH electromagneto-acoustic wavepropagation problem discussed above is examined underthree different sets of boundary conditions.
Problem 1 (The grounded surface solution).In this prob-lem, the surface of the piezoelectric half space~see Fig. 1! iscovered with an infinitesimally thin perfect conducting film,which implies that the additional mechanical effect is ne-glected, and the scalar potential of the electric field is set tobe zero on the surface. This group of boundary conditionsare expressed as follows,
FIG. 1. The piezoelectric half space covered with a layer of conductingfilm.
5265J. Appl. Phys., Vol. 80, No. 9, 1 November 1996 Shaofan Li [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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s23~x1,0,t !5 c44]w
]x21e15
]c
]x21e15cl
]A2
]t50, ~29!
f~x1,0,t !5c1e15e11s fw50. ~30!
Assume
w~x1 ,x2 ,t !5w0 exp@2k1x2#exp@ i ~vt2kx1!#, ~31!
c~x1 ,x2 ,t !5c0 exp@2k2x2#exp@ i ~vt2kx1!#, ~32!
A1~x1 ,x2 ,t !5A10 exp@2k1x2#exp@ i ~vt2kx1!#, ~33!
A2~x1 ,x2 ,t !5A20 exp@2k1x2#exp@ i ~vt2kx1!#, ~34!
where k22k125(v/ca)
2, k22k225(v/cl )
2. The amplitudecoefficientsw0 , c0 , A10, and A20 are not independent.Substitutingw, A1, andA2 into the Lorentz gauge constraint~17!, one may find the following relationships,
A1052m0e15cl vk
~k122k2!1v2/cl
2w0 , ~35!
A205 im0e15cl vk1
~k122k2!1v2/cl
2w0 . ~36!
It should be noted that expressions~35! and ~36! providequantitative evaluation of the electromagneto-acoustic inter-action caused by the piezoelectricity.
By substituting~31!–~34! into the boundary conditions~29!–~30! and by considering the relations~35! and~36!, theboundary conditions~29!–~30! lead to the following homo-geneous algebraic equations,
S S c441 m0e152 v
~k122k2!1v2/cl
2D k1 e15k2
e15e11s f 1
D F w0
c0G50.
~37!
The nontrivial solution exists if and only if
U S c441 m0e152 v
~k122k2!1v2/cl
2D k1 e15k2
e15e11s f 1
U50, ~38!
which yields
k15e15
2
e11s c44
f k2 . ~39!
Let
b2:5e15
2
e11s c44
f5b2f , ~40!
f :5cl
2
cl22b4ca
2, ~41!
where the constant,b:5e15/Ae11s c44, is the piezoelectric
coupling coefficient in the quasi-static approximation. Tothis end, a simple velocity equation for the electromagneto-acoustic surface wave is derived,
ve5caA f ~12b4!. ~42!
As ca /cl →0,
f51
12ca2/cl
2⇒1, ~43!
b25b2f⇒b2, ~44!
f51
12b4ca2/cl
2⇒1; ~45!
the surface wave speed recovers the classic BG solution,
vq5caA12b4. ~46!
Problem 2 (The free surface solution).In this case, thesurface of the piezoelectric half space is a free surface, whichis in contact with a vacuum half space on the top. The full setof boundary conditions atx2 5 0 are as follows,5
n•s50, ~47!
n•D5n•D, ~48!
n3E5n3E. ~49!
Note that the quantities with symbol ^ on the top denote thequantities of the electric field in the free vacuum space.
For this particular boundary configuration, the aboveboundary conditions can be put into explicit forms, i.e., atx250,
c44]w
]x21e15
]c
]x21e15cl
]A2
]t50, ~50!
e15~12 f !]w
]x22e11
s ]c
]x22
e11s
cl
]A2
]t52e0
]f
]x22
e0c0
]A2
]t,
~51!
]c
]x11e15e11s f
]w
]x11
1
cl
]A1
]t5
]f
]x11
1
c0
]A1
]t. ~52!
In addition, one may also note that the boundary conditionfor magnetic induction is always fulfilled, i.e.,
n–B5n–B[0. ~53!
The Maxwell equations in the free vacuum half spacetake the form
¹2f21
c02
]2f
]t250, ~54!
¹2A121
c02
]2A1
]t250, ~55!
¹2A221
c02
]2A2
]t250, ~56!
wherec0 :5 (m0e0)21/2.
5266 J. Appl. Phys., Vol. 80, No. 9, 1 November 1996 Shaofan Li [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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The associated Lorentz gauge in the vacuum space is
]A1
]x11
]A2
]x21
1
c0
]f
]t50. ~57!
Assume
f5f0 exp@k3x2# exp@ i ~vt2kx1!#, ~58!
A15A10 exp@k3x2# exp@ i ~vt2kx1!#, ~59!
A25A20 exp@k3x2# exp@ i ~vt2kx1!#, ~60!
wherek325k22v2/c0
2.To satisfy the Lorentz gauge constraint~57!, the ampli-
tude coefficientsf0 , A10, andA20 are related by
A205i
k3~kA102~v/c0!f0!. ~61!
Substituting ~58!–~60! into the boundary conditions~50!–~52! and considering~61!, one may obtain the follow-ing linear homogeneous equations,
c44k1v01e15k2c050, ~62!
e11s k2k3c01e0k
2@f02~v/c0!A10#50, ~63!
~e15/e11s !v01c02@f02~v/c0!A10#50. ~64!
Eliminate @f0 2 (v/c0)A10# from the above equations. Itthen yields
S c44k1 e15k2
e15/e11s 11~e11
s e0!S k2k3k2 D D F v0
c0G50. ~65!
Letting v 5 v/k and equating the coefficient determinant tozero, one can find that the electromagneto-acoustic surfacewave speed must obey the following velocity equation,
c44e11s A12~ve /ca!
2A12~ve /cl !2A12~ve /c0!2
1e0c44A12~ve /ca!25
e0e11s e15
2 A12~ve /cl !2. ~66!
If v/cl ! 1,v/c0 ! 1, it reduces to
c44~e11s 1e0!A12v2/ca
25e0e11s e15
2 . ~67!
Let
bv2 :5
e152
c44e11s
e0~e11
s 1e0!. ~68!
Once again, we recover the classic result
vq5caA12bv4. ~69!
Remark 3.1.Assumeca,cl ,c0. One can readily verifythat the velocity equation~66! has no real root in the speedrange ca<v<c0. Equation ~66! may have complex rootswithin this speed range, or it may have a real root afterv.c0. Some of these possibilities may imply the existenceof the electromagnetic surface wave on the interface betweenvacuum space and the piezoelectric medium. The quasi-staticapproximation usually fails to predict such possibility.
Problem 3 (the interface solution).This is an analogy ofthe well-known Maerfeld–Tournois surface wave that is alsoobtained under the quasi-static approximation.6 The interfaceis located atx250 between two different piezoelectric halfspaces~see Fig. 2!. In this case, the independent boundaryconditions are as follows:
u5u8, ~70!
n•s5n•s8, ~71!
n•D5n•D8, ~72!
n3E5n3E8. ~73!
One can verify that the magnetic boundary conditions,
n•B5n•B8, ~74!
n3H5n3H8, ~75!
are satisfied automatically.The boundary conditions~70!–~73! can be written ex-
plicitly as
w5w8, ~76!
c44]w
]x21e15
]c
]x21e15cl
]A2
]t5 c448
]w8
]x21e158
]c8
]x2
1e158
cl 8
]A28
]t, ~77!
e15~12 f !]w
]x22e11
s ]c
]x22
e11s
cl
]A2
]t
5e158~12 f 8!]w8
]x22e11
s 8]c8
]x22
e11s 8
cl 8
]A28
]t, ~78!
]c
]x11e15e11s f
]w
]x11
1
cl
]A1
]t
5]c8
]x11e158
e11s 8
f8]w8
]x11
1
cl 8
]A18
]t. ~79!
For the lower half piezoelectric space, we still assumethat
w5w0 exp@2k1x2# exp@ i ~vt2kx1!#, ~80!
c5c0 exp@2k2x2# exp@ i ~vt2kx1!#, ~81!
FIG. 2. An interface between two different piezoelectric media .
5267J. Appl. Phys., Vol. 80, No. 9, 1 November 1996 Shaofan Li [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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A15A10 exp@2k1x2# exp@ i ~vt2kx1!#, ~82!
A25A20 exp@2k1x2# exp@ i ~vt2kx1!#, ~83!
with k22k125(v/ca)
2, k22k225(v/cl )
2; and consequently,
A1052m0e15cl vk
@~k122k2!1v2/cl
2#w0 , ~84!
A205im0e15cl vk1
@~k122k2!1v2/cl
2#w0 . ~85!
For the upper half piezoelectric space, we assume that
w85w08 exp@k18x2# exp@ i ~vt2kx1!#, ~86!
c85c08 exp@k28# exp@ i ~vt2kx1!#, ~87!
A185A108 exp@k18x2# exp@ i ~vt2kx1!#, ~88!
A285A208 exp@k18x2# exp@ i ~vt2kx1!#; ~89!
with k22(k18)25(v/ca8)
2, k22(k28)25(v/cl8 )
2; and conse-quently,
A108 52m0e15cl vk
@~~k18!22k2!1v2/cl2 #w08 , ~90!
A208 52 im0e15cl vk18
@~~k18!22k2!1v2/cl2 #w08 . ~91!
Inserting the assumed solutions~80!–~83! and~86!–~89!into the boundary condition set~76!–~79! and taking equa-tions ~84! and~85! and~90! and~91! into consideration, onemay find the following existence condition for the interfacialelectromagneto-acoustic wave:
U 1 0 21 0
k1c44 e15k2 k18c448 e158 k28
0 e11s k2 0 e11
s8k28
~e15/e11s !k k 2~e158 /e11
s8!k 2k
U50. ~92!
Subsequently, it leads the following wave velocity equa-tion,
c44e11s A12~ve /ca!
2A12~ve /cl !21 c44e11s8A12~ve /ca!
2
3A12~ve /cl 8!21 c448e11s A12~ve /ca8!2
3A12~ve /cl !21 c448e11s8A12~ve /ca8!2
3A12~ve /cl 8!25DA12~ve /cl !2A12~ve /cl 8!2,
~93!
where
D:5F S e15e11s D 2S e158e11
s8 D G2e11s e11s8 . ~94!
When v/cl ,,1, v/cl 8!1, the above velocity equa-tion reduces to the Maerfeld-Tournois wave speed equation,6
c44A12~vq2/ca!
21 c448A12~vq /ca8!2
5F S e15e11s D 2S e158e11
s8 D G2S e11s e11
s8
e11s 1e11
s8D . ~95!
Remark 3.2.Assumeca<ca8 < cl <cl8 . The velocityequation ~93! can only have complex roots in the speedrangeca8<v<cl . After v.cl , it may have real roots. Onceagain, we cannot rule out the possibility that there is apiezoelectrically-induced electromagnetic surface wavepropagating along the interface.
IV. CONCLUSIONS
As shown above, the exact solutions for the fully-coupled SH electromagneto-acoustic surface wave can beobtained in simple, closed forms. As expected, these solu-tions take the classic results, such as BG wave or Maerfeld–Tournois wave, as their limits, when the termsv/cl andv/c0 are negligible.
The new results obtained here reveal the acousto-opticinteraction on the surface of piezoelectric materials, and im-plicitly indicate a possible existence of the piezoelectrically-induced electromagnetic surface wave.
As far as the acoustic wave is concerned, the quasi-staticassumption does offer an excellent approximation for thewave speed. This can be observed from Fig. 3. In Fig. 3, acomparison between the current result and the classic resultis made forProblem 1, the grounded surface solution. Sincein reality the ratioca /cl is within the range of 0;0.0001,the optical effect on acoustic wave speed is minor indeed.
However, the quasi-static approximation may not alwaysmake life easy; it can cause some additional difficulties.Such problems arise when one studies wave propagation orwave scattering problems in a piezoelectric material contain-ing defects.7,8 An obvious reason for this is that the govern-
FIG. 3. The ratiove /vq versusca /cl for the grounded surface solution .
5268 J. Appl. Phys., Vol. 80, No. 9, 1 November 1996 Shaofan Li [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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ing equations obtained under quasi-static approximation aremixed types of partial differential equations; namely, one ishyperbolic, the other is elliptic, because the speed of light isassumed to be infinite in the quasi-static approximation. Byusing the approach proposed in this paper, one may obtainequally simple decoupled governing equations, and they areboth hyperbolic type of partial differential equations.
It should be reminded that all the surface waves studiedhere are nondispersive. There may exist dispersiveelectromagneto-acoustic surface wave mode in some piezo-electric solids with particular geometric shapes, such as aninfinite piezoelectric plate. Under those circumstances, theremay be a nontrivial discrepancy for the high frequency sur-face wave solutions between the approach proposed in thispaper and the quasi-static approximation. This issue is sub-jected the further study.
ACKNOWLEDGMENTS
This research is supported by a grant from Naval Re-search Office. The author is grateful for the referee’s con-structive comments.
1J. L. Bleustein, Appl. Phys. Lett.13, 412 ~1968!.2Y. V. Gulyaev, Sov. Phys. JETP9, 37 ~1969!.3B. A. Auld, Acoustic Field and Wave in Solids~Wiley, New York, 1990!,Vol. 1.
4B. A. Auld, Acoustic Field and Wave in Solids~Wiley, New York, 1990!,Vol. 2.
5J. D. Jackson,Classical Electrodynamics, 2nd ed.~Wiley, New York,1975!.
6C. Maerfeld and P. Tournois, Appl. Phys. Lett.19, 117 ~1971!.7S. Li and P. A. Mataga, J. Mech. Phys. Solids~to be published!.8S. Li and P. A. Mataga, J. Mech. Phys. Solids~to be published!.
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